Question

For a complex number $Z$ , if the argument of $3+3i$ and $\left(Z-2\right)\left(\stackrel{-}{Z}-1\right)$ are equal, then the maximum distance of $Z$ from the $x$ -axis is equal to (where, $i^2=-1$ )

• A. $\frac{\left(1+\sqrt{2}\right)}{2}$ units
• B. $2$ units
• C. $\frac{3}{2}$ units
• D. $\frac{\left(\sqrt{2}+2\right)}{2}$ units

Answer 1

Answer: (A)

Let, $Z=x+iy$
Now, $\left(Z-2\right)\left(\stackrel{-}{Z}-1\right)=Z\stackrel{-}{Z}-2\stackrel{-}{Z}-Z+2$
$=x^2+y^2-3x+2+iy$
$arg$ of $\left(3+3i\right)=\frac{\pi }{4}$
$\Rightarrow x^2+y^2-3x+2=y\left(wherey>0\right)$
$\Rightarrow x^2+y^2-3x-y+2=0$
Centre of the circle is $\left(\frac{3}{2},\frac{1}{2}\right)$ and radius $=\frac{1}{\sqrt{2}}$

$\Rightarrow$ required value $=\frac{1}{2}+\frac{1}{\sqrt{2}}$ units